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KANTS THEORY OF SPACE AND THE NON-EUCLIDEAN GEOMETRIES
In the transcendental exposition of the concept of space in the “Space” section of the
Transcendental Aesthetic Kant argues that “geometry is a science which determines the
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properties of space synthetically and yet a priori” . Together with the claims from the
metaphysical exposition in the same section that space is not derived from any outer
experience but it is a pure intuition and necessary a priori representation that is given as
infinite magnitude, this builds up the general framework of the relation between space
and geometry in Critique of Pure Reason. For Kant there exists only one geometry and
this is the Euclidean geometry. On this basis runs what Friedman calls “the standard
modern complaint against Kant”, namely, that he did not make the crucial distinction
between pure geometry and applied geometry. Since pure geometry makes no appeal to
spatial intuition or other experience and since the truth of the axioms of the applied
geometry depends upon an interpretation in the physical world the question about which
axiomatic system, the pure or the applied one, is true is settled only by empirical
investigation2. This directly contradicts Kants fundamental claim that we can know the
proposition of the Euclidean geometry a priori.
Major part in this complaint is played by appeal to the non-Euclidean geometries.
Historically, this was initiated by Helmholtz who argued that Kants theory of space is
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untenable in the light of the discovery of the non-Euclidean geometries . His line was
later forcefully supported by Paton, Russel, Carnap, Schlick and probably at most
Reichenbach, who famously criticized Kants conception of space on the basis of a
complex analysis of the visual a priori which he took to underlie Kants doctrine of
geometry4. Recently, Parsons refers to this line as “the most common objections to
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Kants theory of space” and concedes that Kantian could still accept some more
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primitive geometrical properties (than those provided by the 5 postulate of Euclids
Elements, for example) to be known a priori even if he abandons the claim that in
specific propositions of the Euclidean geometry can be known a priori. Though this is an
attempt to salvage some part of the geometry doctrine I do not think that this is in the
spirit of the Transcendental Aesthetic and also, I believe that it would be insufficient for
Kants purposes.
My aim in this paper will be defend the view that Kants doctrine of geometry can
survive criticism based on appeal to the non-Euclidean geometries. I will argue that we
can still make sense of Kants claim that it is the Euclidean geometry that determines the
properties of space and that it does it a priori provided that we have proper understanding
of his space conception as a pure form of the intuition. I will try to show that Kants
appeal to basic propositions of the Euclidean geometry as necessary true and intuitively
certain could still be defended. I will argue, partly in the sense of Friedman, that we do
not have to accept the pure – applied geometry distinction but my justification will not be
based on his interpretation of the connection between Kants logic and philosophy of
mathematics. I would rather argue that such distinction is inappropriately applied ad hoc
to the doctrine of space alone but its claim is of much bigger scope and as such should be
met by frontal debate against Kants system of transcendental idealism. Such debate,
however, often is either not offered by the critics or dissolves the strength of the criticism
upon much broader field.
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If the distinction is dispensed in such way, most critiques based on appeal to the
non-Euclidean geometries will be reduced to the principle claim that the propositions of
the Euclidean geometry are not the only candidates for true with certainty any more but
the propositions of the non-Euclidean geometry are competitors as well. The common
reading of this claim is that the non-Euclidean propositions are also intuitable or
visualizable in our intuition, as such they could (alone or also) be true of it and therefore
the Euclidean ones are not apodeictic, necessary true and intuitively certain. I will claim
that the non-Euclidean propositions cannot be true of the space of the intuition since they
lack a set of substantial properties that are necessary for the space of the intuition and that
are actually found there upon introspection. I will argue that only the Euclidean geometry
provides these properties and as such it is indispensable for the space of the intuition not
only in the Kants but in a broader sense.
PURE – APPLIED GEOMETRY DISTINCTION
Among the recent studies of the topic probably Friedman formulated most clearly this
objection, when he says that “Kant fails to make the crucial distinction between pure and
applied geometry”6. Interestingly, this approach has been used in both directions, as
criticism of Kants theory of space (Russell, Reichenbach, and Hopkins) but also as a
way to save the doctrine (Ewing, Strawson). The division gave birth to different
interpretations that multiplied the possible options among which we could choose from.
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Thus Schiller divided the space to perceptual and conceptual , Craig proposed three
readings of what the geometrical axioms (the Euclidean ones) could be true of, the space
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of sense-impressions, the space of mental images and the space of the way things look .
Lucas distinguished between four different senses of use of the word “space”: as a term
of the pure mathematics, as a term of the physics, as a space of our ordinary experience
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and as a possibility for existence (construction) of objects (geometrical and physical) .
Satisfactory comment on all these readings is beyond the scope of the present discussion
but it is important to point that although all they have merits of their own yet some of
them fail to account properly for one but crucial aspect of spaces meaning in the
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Critique of Pure Reason, namely that space is pure intuition and was never meant to
apply to things in themselves. Traditionally, this is the sense in which applied geometry
is interpreted and at present day most physicist regard the rules of geometry, whatever
they are as applied to the realm of the things as they are by themselves. Pure intuition,
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however, is what provides for Kant immediate representation to objects for the subject
and which not only precedes the actual appearances of the objects but in fact makes them
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possible . The term “pure” is clarified (in the transcendental sense) as “there is nothing
that belongs to sensation”.13 Together with the claim that “space does not represent any
property of things in themselves, nor does it represent them in their relation to one
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another” this implies that on the one hand the space notion Kant builds is not
determined by the experience but on the opposite, makes it possible and on the other hand
that has no claim whatsoever about the world of the things in themselves, what at present
day is meant under the term “physical world”.
Proper understanding should recognize that the lack of epistemic access to the
world of the things in themselves is not simply due to the character of the space notion
but it is a fundamental feature of Kants epistemic and ontological model of the world.
Taking this into account any criticism against the non-applicability of the space to the
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physical world rush ahead and neglect the fact that such ambitious criticism is to be
targeted against the complete model instead. Since it is not that much of a criticism
against space that it is not applied to the physical world, the responsibility for the
application of space is dependent on the general model. In this sense it is not a deficiency
of Kants doctrine of space and geometry in particular whether it is or it could be true of
the physical world. The space notion and the role of geometry is to be understood
properly only within Kants system and within the system the notion of space is
consistent and even necessary for the lack of appeal to the physical world as such (an
sich). Any criticism against the big model, however, was rarely if at all presented as
accompanying talks about space and geometry. In this sense, any objection of the
simplified form “Kant claims that the propositions of the Euclidean geometry are the only
true ones as describing space, our space is the physical space and not a subjective one,
there is physical geometry that actually describes the space truly and this geometry is
non-Euclidean, therefore Kants theory of space and geometry is wrong” is without the
necessary back up, namely, the very reason why Kant is wrong when he says that we do
not know the things in themselves but only as they appear to us. Such criticism, however,
becomes much more complicated than the relatively simple discussion about space and
geometry and since it would position itself within centuries of unresolved philosophical
debates it is much less obvious than the ad hoc objections about geometry. It is not
surprising that most critics prefer to attack the concrete issue and avoid more
fundamental debate. However, no such approach could be entirely successful since it
does not explain why we should prefer, for example, direct realism about the external
world instead a representational theory of the kind proposed by Kant. I believe that mere
appeal for skeptical implausibility about external world as feature of Kants general
model will not do the job, at least with respect to space and geometry. Without successful
final of frontal criticism against the general model there is no justification for introducing
“applied geometry” as an option of a distinction as it is meant to apply to the world of the
things in themselves at all. This would, however, undermine the reasons for introducing
the distinction with respect to Kants theory of space and geometry.
As an illustration, often the objection about the non-applicability of geometry to
the physical world takes the following form, the example here being taken from Russell:
On the other hand there is geometry as a branch of physics, as it appears, for example, in the general theory
of relativity; this is an empirical science, in which the axioms are inferred from measurements and are
found to differ from Euclids. Thus …. it is synthetic but not a priori15.
There are two difficulties for Kant here: one about the a posteriori character of the
axioms and another (an implicit one) that the Euclidean geometry is not true as applied to
the physical world. Response to this could be the following: even if the large scale of the
universe is properly described according to the general theory of relativity by a spherical
geometry the scale of the ordinary human experience is still almost complete
approximation of the Euclidean geometry. The differences are so minute that they are
practically undistinguishable. In this sense the Euclidean geometry is still true when
applied to some scales and definitely true when applied to the human scale ordinary
experience. Further, for Kant the question would not be that much if the geometry is
applied to the world of the things in themselves since this option is ruled out in general
but whether this geometry is true of how the world appears to us. Here, the supposition
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that this is a non-Euclidean geometry is not obvious at all. Later in the paper I will argue
that practically the geometry of the space of the intuition is Euclidean and cannot be non-
Euclidean. Nevertheless all this is more or less irrelevant for Kants space doctrine since
the physical geometry and the pure one would coincide in his system. There will be only
one geometry in and this would be the geometry that reigns the space of the intuition.
This is Strawsons point in The Bounds of Sense when he comments on the traditional
criticism:
He thought that the geometry of the physical space had to be identical with the geometry of the
phenomenal space. And this mystery does invite the suggestion that the geometry of the phenomenal space
embodies, as it were, conditions under which alone things can count as things in space, as physical objects,
for us. Especially does it invite this suggestion if we think of somethings counting as a physical body for
us in terms of its appearing to us, presenting to a phenomenal figure …16
This geometry is not related to the world of the things in themselves at all, it prescribes
(geometrical) predicates only as far as the objects appear to us.17 Also, even if this
geometry happens to be non-Euclidean one this still does not mean that it is a posteriori
since, as Jones points
Nor did Gauss, Lobachevski, or Bolyai follow such (empirical, my note) procedure in developing non-
Euclidean geometry. They all carried out their work without recourse to experience, and thus a priori. Just
what their criterions of truth need not be considered here, but they certainly were not empirical …18
So, after the claim that geometry must describe the physical world as such is dismissed
the argument against apriority of the axioms could be met by simple appeal to the history
of the development of the non-Euclidean geometries and the foundations of their axioms.
A weaker variation of criticism based on non-Euclidean geometries says that since Kant
affirmed that only one geometry is true of the intuitive space the very discovery of the
alternative geometries proved him wrong. This could be met by pointing that Kant
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actually anticipated such possibilities and that the question is reduced to the next one
whether the geometries are and could be intuitively true. As I mentioned above, this
question will be discussed further in the paper.
To sum up, an attempt to criticize Kants theory of space on the basis that he did
not distinguished between pure and applied geometry could not be successful since the
question of what does the space apply to with Kant is resolved by appeal to the world as
it appears to us. Whatever this world looks like, certain geometry is applied to it and the
question why this geometry is not applied to the world of the things in themselves is
meaningless for Kant since the only thing we can know about such world is that it exists
and no geometrical predicate can be applied to it or to its objects and the relations
between them. The further question about the status of the non-Euclidean geometries is
thus reduced to the question whether they apply to the space of the intuition and how. In
addition, an important explanatory remark in this respect is the point made by Friedman
who argues convincingly that the distinction between pure and applied geometry goes
together with certain understanding of logic that was not available to Kant since appears
with Gottlobs Frege Begriffsschrift in 1879.20 The importance of this relation is, as
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Friedman stresses, also supported by Hintikka and Parsons .
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